Figure 1. Schematic of the Mach-Zehnder interferometer used in the quantum eraser experiments. One additional polariser is required between the source and the quarter wave plate as indicated in the figure.
Aim:
To examine key elements of which-way measurement, quantum erasure, and perceived mysteries of quantum measurement.
Introduction:
The mysteries of complementarity and the measurement process in quantum mechanics are perhaps most clearly demonstrated in the context of interference. Interference of light is normally explained in the context of the superposition of light waves. When a wave can travel form a source to a detection region, interference fringes can be seen on a detector, which may be just a screen, that depend on the relative phase of the waves that are combining to form the interference pattern. In a quantum mechanical description, when a photon can travel from a source to a detection region via two distinctly different paths, interference between the two paths can be detected by interference fringes, which give the spatial dependence of the photon probability distribution. The important point is that the photon is considered to have travelled both paths and the photon is interfering with itself to produce the fringe pattern. You should use this concept to describe the experiments you will do in this lab. It is also possible to describe the experiments classically, in terms of interfering waves. Provided the two waves have a common polarisation component interference is possible. The classical model breaks down when the light intensity is so low that only a single photon at a time can be in the interferometer. Then only the quantum explanation works. You will not be operating in this regime today, but you can still contemplate the quantum explanation of the effects you observe.
Which way?
This is a key question in the quantum explanation. In the quantum picture, interference only occurs if we have no way of determining which path the photon took from the source to the detection region. If we set up the experiment in such a way that we could establish which path the photon took, then the interference fringes disappear. In fact we do not actually have to establish which path the detected photon did take; just the possibility of being able to ascertain the path, for example by placing a polariser before the detector, is sufficient to destroy the interference.
Experimental arrangement:
We use a Mach-Zehnder interferometer. A basic Mach-Zehnder interferometer is shown diagrammatically in Figure 1. In our setup, the source is a helium neon laser. At the detectors, which in our case are just screens, interference fringes should be visible using the arrangement of Figure 1. Ideally the arms of the interferometer should be of equal length, but this is not critical as long as the coherence length of the laser exceeds the difference in the path length $L_c=c/(\pi \Delta f)$. A typical He-Ne laser has a wavelength of 632 nm and spectral width of about 800 MHz.
Set up the interferometer on the optical breadboard. Most mounts for the components are in place, but it will still be necessary to insert the correct components in the correct mounts and align the beams carefully. Initially, leave out the lens and all polarizers after the lens. It is easier to initially align without the lens.
Polarization is crucial in this experiment. Install a linear polariser immediately after the laser ("add polariser") so that its longitudinal modes become polarized at 45° to the vertical. Install the $\lambda/4$ plate with its optic axis vertical. This converts the linearly polarised beam to circular polarisation, ensuring equal amounts of vertical and horizontal polarisation in both arms of the interferometer.
For beamsplitter BS1 use the flat plate 50/50 beamsplitter. For beamsplitter BS2, use the non-polarising cube beamsplitter. It is easier to align the interferometer that way. Some useful tips for aligning: It is much easier to obtain fringes by keeping all the beams in the interferometer at the same height. Measure the height of the beam at mirror 1 in the diagram, and adjust mirror 1 vertically so the beam stays at that height. After installing BS1 make sure the height at mirror 3 is the same, and after installing mirrors 2 and 3, check the height at BS2. To combine the beams at BS2, put a business card immediately behind the beamsplitter and adjust mirrors 2 and 3 to bring the two spots together. To overlap the spots far away from the beamsplitter, adjust only the beamsplitter, not the mirrors. You will need to repeat the process a few times to get the beams overlapped everywhere. Patience is essential.
Observations:
When the spots are aligned, you should see a fringe pattern. You may need to look carefully as they may be close together. If you happen to have everything perfectly aligned, the fringes will be circular. Most likely they will be straight instead, indicating slight misalignment, but it is OK and even preferable to work with the straight fringes.
When you see the interference pattern, please contact the instructor who will verify the interferometer is correctly aligned before you continue further.
You should now insert the lens. Adjust only the lens until you get the fringes back. Make sure the beam goes through the centre of the lens (vertical and horizontal positioning of the lens may be required), and look at the fringe patterns while you make the final adjustment. The fringes will now be bigger and easier to see.
Block half of the beam in one of the arms of the interferometer and note the effect on the fringe pattern. Explain in terms of the quantum mechanical picture. We now attempt to identify which path a photon has travelled. We can do this by placing polarizers in each path. Initially orient both polarisers vertically.
Now rotate one of the polarisers so that it is horizontal.
One way that the path of the detected photon could be determined is by placing a vertical or horizontal polariser after BS2 and before the screen. Explain how this determines the which way information. Now rotate the polariser between BS2 and the screen so that it is 45°. This is the quantum eraser setup. This polariser is said to be erasing the potential to ascertain which way information.
Image the fringe pattern with the polariser at 45°. Now rotate the polariser through 90° so it is at 45° the other way. Note any changes in the fringe pattern. For this exercise, the bigger the fringes are the better. As a final experiment, place a vertical polarizer in one of the arms of the interferometer, with no polarizer in the other arm. Place a horizontal polarizer in one of the output beams, with no polarizer in the other output beam.
Laser light may be modeled as a plane wave propagating in the z-direction having an angular frequency $\omega$ and wave number $k=\omega/c$, and transverse electric field components $ {\begin{pmatrix} E_{x}(t)\\ E_{y}(t)\end{pmatrix}} ={\begin{pmatrix}E_{0x}e^{i(kz-\omega t+\phi _{x})}\\E_{0y}e^{i(kz-\omega t+\phi _{y})}\end{pmatrix}}={\begin{pmatrix}E_{0x}e^{i\phi _{x}}\\E_{0y}e^{i\phi _{y}}\end{pmatrix}}e^{i(kz-\omega t)}. $ The amplitude $ {\begin{pmatrix}E_{0x}e^{i\phi _{x}}\\E_{0y}e^{i\phi _{y}}\end{pmatrix}} $ of this wave is called the Jones vector and it represents the magnitude and phase of the electric field in the x and y directions.
Polarisation state of polarised light can conveniently be modeled using dimensionless Jones vectors. A few such Jones vectors are:
Linear polarized in the x "horizontal" direction: \begin{equation*} \rm{LH} = \begin{bmatrix} 1 \\ 0 \\ \end{bmatrix} \end{equation*}
Linear polarized in the y "vertical" direction: \begin{equation*} \rm{LV} = \begin{bmatrix} 0 \\ 1 \\ \end{bmatrix} \end{equation*}
Left circularly polarized: \begin{equation*} \rm{CL} = \frac{1}{\sqrt{2}}\begin{bmatrix} 1 \\ i \\ \end{bmatrix} \end{equation*}
Right circularly polarized: \begin{equation} \rm{CR} = \frac{1}{\sqrt{2}}\begin{bmatrix} 1 \\ -i \\ \end{bmatrix} \end{equation}
Action of polarising optical elements can be conveniently modeled using Jones matrices. A few such Jones matrices are:
Linear polarizer with axis of transmission horizontal: \begin{equation*} \rm{LPH} = \begin{bmatrix} 1 & 0 \\ 0 & 0\\ \end{bmatrix} \end{equation*}
Linear polarizer with axis of transmission vertical: \begin{equation*} \rm{LPV} = \begin{bmatrix} 0 & 0 \\ 0 & 1\\ \end{bmatrix} \end{equation*}
Linear polarizer with axis $\theta$ from the horizontal: \begin{equation*} \rm{LPW} = \begin{bmatrix} \cos^2(\theta) & \cos(\theta) \sin(\theta) \\ \cos(\theta) \sin(\theta) & \sin^2(\theta) \\ \end{bmatrix} \end{equation*}
Quarter wave plate: \begin{equation*} \rm{QWP} = e^{\frac{i\pi}{4}}\begin{bmatrix} 1 & 0 \\ 0 & -i \\ \end{bmatrix} \end{equation*}
Python implementation of the states (vectors) and operators (matrices) looks like:
import numpy as np
LH = np.array([1,0])
LV = np.array([0,1])
CL = np.array([1,1j]) / np.sqrt(2)
CR = np.array([1,-1j]) / np.sqrt(2)
LPH = np.array([[1,0],
[0,0]])
LPV = np.array([[0,0],
[0,1]])
import pypolar.visualization as vis
vis.draw_jones_animated(CR,nframes = 10)
beam1 = LPH @ CR
vis.draw_jones_animated(beam1,nframes = 10)
/Users/narmolaksingh/anaconda3/lib/python3.11/site-packages/matplotlib/animation.py:884: UserWarning: Animation was deleted without rendering anything. This is most likely not intended. To prevent deletion, assign the Animation to a variable, e.g. `anim`, that exists until you output the Animation using `plt.show()` or `anim.save()`. warnings.warn(
The polarisation state of the light (Jones vectors) can be conveniently visulised using the pypolar package:
!pip install pypolar
import pypolar.visualization as vis
vis.draw_jones_animated(CR,nframes = 10)
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When a beam goes through an optical element we simply left multiply the Jones vector of the beam by the Jones matrix of the optical element to obtain the new polarisation state of the light after it has passed through the optical element:
# Compute the polarization state of the initially right circularly polarised light after passing it through a horizontal linear polarizer
beam1 = LPH @ CR
vis.draw_jones_animated(beam1,nframes = 10)
In all cases, in addition to the code, show the Python prediction (image) for the polarisation states of the two beams (first week) and upload a photo of the experimental measurement (second week), and provide a written explanation for your observations. Use the layout shown below (or similar).
# Import necessary libraries
import numpy as np
import pypolar.visualization as vis
# Jones vector for Beam 1 (e.g., Left Circular Polarization)
beam1 = np.array([1, 1j]) / np.sqrt(2)
# Jones vector for Beam 2 (e.g., Right Circular Polarization)
beam2 = np.array([1, -1j]) / np.sqrt(2)
# Visualize the polarization state of Beam 1
vis.draw_jones_animated(beam1, nframes=10)
# Visualize the polarization state of Beam 2
vis.draw_jones_animated(beam2, nframes=10)
/Users/narmolaksingh/anaconda3/lib/python3.11/site-packages/matplotlib/animation.py:884: UserWarning: Animation was deleted without rendering anything. This is most likely not intended. To prevent deletion, assign the Animation to a variable, e.g. `anim`, that exists until you output the Animation using `plt.show()` or `anim.save()`. warnings.warn(
# Superposition of Beam 1 and Beam 2 (overlapping beams)
combined_beam = beam1 + beam2
# Visualize the polarization state of the combined beams
vis.draw_jones_animated(combined_beam, nframes=10)
Answer:
The coherence length is given by:
$L_c = \frac{c}{\pi \Delta f}$
where:
Given:
Calculations:
calculating the denominator ($\pi\Delta f$)
$\pi\Delta f = \pi \times 800 \times 10^6 \text{ Hz} = 2,513,274,123 \text{ Hz}$
the coherence length ($L_c$)
$L_c = \frac{c}{\pi\Delta f} = \frac{3 \times 10^8 \text{ m/s}}{2,513,274,123 \text{ Hz}} \approx 0.1194 \text{ m}$
So, $L_c \approx 0.1194 \text{ m}$ (approximately 12 centimeters).
Therefore, the paths in the interferometer can be unequal by up to approximately 12 centimeters before the interference fringes are lost.
*Answer:
In the classical picture, light is described as a wave phenomenon. The interference fringe pattern arises due to the superposition of two coherent light waves traveling along different paths in the interferometer and recombining at the beam splitter. When these waves meet, they interfere constructively or destructively depending on the phase difference between them at each point on the screen or detector.
The interference pattern is thus a result of the spatial variation of the intensity due to the varying phase difference caused by differences in optical path lengths or angles between the two beams.
Answer:
In quantum mechanics, the fringe pattern arises due to the superposition of a single photon traveling both paths simultaneously. The photon interferes with itself at the detector, and as long as no "which-way" information is available (i.e., we don't know through which path the photon traveled), interference fringes appear. Once the which-way information becomes available, the interference is destroyed, and the fringes disappear.
In a Mach-Zehnder interferometer, interference occurs when no which-way information is available. When beams are coherent and aligned, interference fringes appear due to the constructive and destructive interference of photons.
The experimental image of the overlapping beams 1 and 2 looks like:
Explanation: Answer:
we should be able to see a fringe pattern if the paths are aligned and no which-way information is present. The images of Beam 1 and Beam 2 should show circular or elliptical polarization states (depending on how they are manipulated), and the experimental image should show the resulting interference.
Theoretical Polarization State of Beam 1 Circular polarization represented by the Jones vector: \begin{equation*} \rm{Beam\ 1} = \frac{1}{\sqrt{2}} \begin{bmatrix} 1 \\ i \end{bmatrix} \end{equation*}
Theoretical Polarization State of Beam 2 Opposite circular polarization represented by the Jones vector: \begin{equation*} \rm{Beam\ 2} = \frac{1}{\sqrt{2}} \begin{bmatrix} 1 \\ -i \end{bmatrix} \end{equation*}
Explanation: In this case, the experiment matches the theory, as no which-way information is available, allowing the photons to interfere with themselves and produce the fringe pattern.
Polarisation state of the theoretical output beam 1 looks like:
Refer to the animation of beam 1 in the cell below.
Polarisation state of the theoretical output beam 2 looks like:
Refer to the animation of beam 2 in the cell below the beam 1 animation.
The experimental image of the overlapping beams 1 and 2 looks like:
Explanation:
Yes, we can clearly see a fringe pattern in the experimental image. This agrees with the theoretical prediction for the following reasons:
Theoretical prediction: The polarisation states of beams 1 and 2 should be circular or elliptical, allowing for interference when they overlap.
Experimental observation: The image shows a clear interference pattern with alternating bright and dark fringes, indicating that the two beams are indeed interfering.
Agreement with theory: The presence of the fringe pattern suggests that: a) The paths of the two beams are aligned properly. b) There is no "which-way" information available, allowing the photons to interfere with themselves. c) The coherence of the laser light is maintained through both paths.
Quantum interpretation: The fringe pattern demonstrates the wave-like nature of light in this experiment. Each photon is in a superposition of states, having traveled both paths simultaneously, and interferes with itself at the detector.
Classical interpretation: The interference pattern results from the superposition of two coherent light waves, with constructive interference creating bright fringes and destructive interference creating dark fringes.
The experiment agrees well with the theory because we observe the expected interference pattern, confirming that the experimental setup maintains the conditions necessary for quantum interference to occur. This result supports both the wave nature of light in classical optics and the principle of superposition in quantum mechanics.
# Import necessary libraries
import numpy as np
import pypolar.visualization as vis
# Define Jones vectors for Beam 1 (Left Circular Polarization) and Beam 2 (Right Circular Polarization)
beam1 = np.array([1, 1j]) / np.sqrt(2) # Left circular polarization
beam2 = np.array([1, -1j]) / np.sqrt(2) # Right circular polarization
# Visualize polarization state of Beam 1
vis.draw_jones_animated(beam1, nframes=10)
# Cell 2: Visualize theoretical polarization state of Beam 2
vis.draw_jones_animated(beam2, nframes=10)
# Combined interference pattern of overlapping beams
combined_beam = beam1 + beam2
vis.draw_jones_animated(combined_beam, nframes=10)
Polarisation state of the theoretical output beam 1 looks like: Refer to the animation in the cell below this.
Polarisation state of the theoretical output beam 2 looks like:Refer to the animation in the cell below the cell showing animation of beam 1.
The experimental image of the overlapping beams 1 and 2 looks like:
Explanation: The experimental observation agrees with the theoretical prediction for the following reasons:
Theoretical Prediction: The polarization states of beams 1 and 2 should be circular or elliptical, allowing for interference when they overlap.
Experimental Observation: The observed image shows an interference pattern with alternating bright and dark fringes, which is characteristic of two coherent beams interfering.
Agreement with Theory: The presence of the fringe pattern suggests that:
Quantum Interpretation: The fringe pattern illustrates the wave-like nature of light in this experiment. Each photon is in a superposition of states, having traveled both paths simultaneously, and interferes with itself at the detector.
Classical Interpretation: The interference pattern arises from the superposition of two coherent light waves. Constructive interference creates bright fringes, while destructive interference results in dark fringes.
# Define horizontal polarizer and apply to Beam 1
horizontal_polarizer = np.array([[1, 0], [0, 0]])
beam1_horizontal = horizontal_polarizer @ beam1
# Visualize polarization state of Beam 1 after horizontal polarizer
vis.draw_jones_animated(beam1_horizontal, nframes=10)
# Visualize polarization state of Beam 2 (unchanged)
vis.draw_jones_animated(beam2, nframes=10)
# Combined pattern with horizontal polarizer on Beam 1 (no interference expected)
combined_no_interference = beam1_horizontal + beam2
vis.draw_jones_animated(combined_no_interference, nframes=10)
Polarisation state of the theoretical output beam 1 looks like: Refer to the animation in the cell below.
Polarisation state of the theoretical output beam 2 looks like: Refer to the cell below the cell showing the animation for beam 1.
The experimental image of the overlapping beams 1 and 2 looks like: 
Explanation: Before Polarizer: we Expect a fringe pattern if the paths are coherent and no which-way information is available.
Explanation:
Classical Picture: The polarizer blocks certain polarization components, which reduces or eliminates the interference fringes.
Quantum Picture: If the polarizer is at 45°, it erases the which-way information, allowing the fringes to reappear.
# Define 45° polarizer and apply to Beam 1
theta = np.pi / 4
polarizer_45 = np.array([[np.cos(theta)**2, np.cos(theta) * np.sin(theta)],
[np.cos(theta) * np.sin(theta), np.sin(theta)**2]])
beam1_45 = polarizer_45 @ beam1
# Visualize polarization state of Beam 1 after 45° polarizer
vis.draw_jones_animated(beam1_45, nframes=10)
# Visualize polarization state of Beam 2 (unchanged)
vis.draw_jones_animated(beam2, nframes=10)
# Combined interference pattern with 45° polarizer on Beam 1 (interference restored)
combined_eraser = beam1_45 + beam2
vis.draw_jones_animated(combined_eraser, nframes=10)
Polarisation state of the theoretical output beam 1 looks like: Please refer to the animation of the cell below.
Polarisation state of the theoretical output beam 2 looks like: Please refer to the animation by the cell below showing the animation for beam 1.
The experimental image of the overlapping beams 1 and 2 looks like:
Explanation:
In the quantum picture, without a polarizer, there is no way to determine the photon's path, so the interference fringes should appear. This supports the idea that superposition and coherence are maintained without which-way information.
# Visualize polarization state of Beam 1 (original circular polarization)
vis.draw_jones_animated(beam1, nframes=10)
# Visualize polarization state of Beam 2 (original circular polarization)
vis.draw_jones_animated(beam2, nframes=10)
# Combined interference pattern without any polarizer
combined_full_interference = beam1 + beam2
vis.draw_jones_animated(combined_full_interference, nframes=10)
Polarisation state of the theoretical output beam 1 looks like:Please refer to the cell animation below. Polarisation state of the theoretical output beam 2 looks like:Please refer to the cell animation below.
The experimental image of the overlapping beams 1 and 2 looks like:(before polariser, after polariser)
After polariser (which way information not known):
Explanation:
Before Polarizer: Expect to see a fringe pattern if no which-way information is available.
After Polarizer:
Explanation:
As you rotate the polarizer through 360°, the interference pattern will change. At 0° and 90°, the which-way information prevents interference, but at intermediate angles like 45°, the which-way information is erased, and the fringes reappear.
# Define Jones vectors for Beam 1 (Left Circular Polarization) and Beam 2 (Right Circular Polarization)
beam1 = np.array([1, 1j]) / np.sqrt(2)
beam2 = np.array([1, -1j]) / np.sqrt(2)
# Visualization of initial polarization states of Beam 1 and Beam 2
vis.draw_jones_animated(beam1, nframes=10)
vis.draw_jones_animated(beam2, nframes=10)
# Define initial beam states
beam1 = np.array([1, 1j]) / np.sqrt(2) # Left circular polarization
beam2 = np.array([1, -1j]) / np.sqrt(2) # Right circular polarization
# Pattern before polarizer (superposition of both beams)
print("Pattern BEFORE polarizer:")
combined_beam = beam1 + beam2
vis.draw_jones_animated(combined_beam, nframes=10)
Pattern BEFORE polarizer:
# Import necessary libraries
import numpy as np
import pypolar.visualization as vis
# Define Jones vectors for Beam 1 and Beam 2 (left and right circular polarization)
beam1 = np.array([1, 1j]) / np.sqrt(2) # Left circular polarization
beam2 = np.array([1, -1j]) / np.sqrt(2) # Right circular polarization
# 1. Interference pattern without any polarizer (before polarizer)
combined_beam_before = beam1 + beam2
print("Interference Pattern Before Polarizer (Clear Fringes):")
vis.draw_jones_animated(combined_beam_before, nframes=10)
Interference Pattern Before Polarizer (Clear Fringes):
# 2. Define a horizontal polarizer (Jones matrix for horizontal polarization)
horizontal_polarizer = np.array([[1, 0], [0, 0]])
# Apply the horizontal polarizer to Beam 1, which introduces which-way information
beam1_with_polarizer = horizontal_polarizer @ beam1
# Recombine the beams after applying the polarizer to one beam
combined_beam_after = beam1_with_polarizer + beam2
print("Interference Pattern After Applying Horizontal Polarizer (Reduced Fringes):")
vis.draw_jones_animated(combined_beam_after, nframes=10)
Interference Pattern After Applying Horizontal Polarizer (Reduced Fringes):